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How do we interpret a data matrix, $X$ which does not have $1s$ as the first column? Does it refer to No Intercept Form?

Could it also be interpreted as the mean deviated form? I understand that for mean deviation form, $({X'X})^{-1}$ matrix has order $k-1*k-1$, but I cannot seem to derive the mean deviated form of $(X'y)$ which has order $k-1*1$. Please help.

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  • $\begingroup$ It could have an intercept lurking within its variables. For instance, the model with matrix $$X_0=\pmatrix{0 & 1 \\ 0 & 1 \\ 1 & 0 \\ 1 & 0}$$ is equivalent to that with matrix $$X_1=\pmatrix{1 & 1 \\ 1 & 1 \\ 1 & -1 \\ 1 & -1}.$$ The first is a "dummy coding" of a binary explanatory variable while the second, whose first column is a constant $1,$ is an "effects coding" of the same variable. $\endgroup$
    – whuber ♦
    Commented Nov 26, 2018 at 14:58

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The column of 1's is added to the data matrix. See e.g. here or here or, if both of those somehow vanish, Google "regression matrix" for more sites.

The 1's are added so that there will be an intercept, and a $\beta_0$ term is in the regressors to estimate that term.

If you wanted to do regression without an intercept (not generally recommended) then you would leave off the column of 1's and $\beta_0$.

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